How do I control scenes switching
Because you are using an actual video, Blender doesn't know when it's finished. At least not if you use a plane and a video texture. In that case, as soon as the scene where that video is located gets activated, the video starts playing.

I program all everything that's scene-related to the camera, since that is an easy object to find and it isn't used to program so it's a clean space. Here I connected a 'scene' actuator to whatever setup you use to click the 'play'button. It should be set to 'set scene' and include the scene you want to activate (which in this case is your video scene).


So now, once we click 'play', we should get into the video scene and the video begins. Since Blender doesn't know when it ends, we should know. Therefore you should take a look at how long the video is (in seconds). Then keep in mind that the refresh rate of your game is by default set to 60. This means ever 'tick' in your game is a refresh. We can use this to let Blender know exactly when to switch over to the next scene.

Here I used an always sensor with a positive pulse trigger set at a skip of 0. This means that every time when the game refreshes (60 times per second) the always sensor sends a true signal to the controller. Now we add a game Property. Give it a name and set it to 'integer' mode. Make sure it's set to 0. On the always sensor we added, add an 'and' controller and after that, a 'Property' actuator. Set the mode of the actuator to 'Add' and set the value to 1. We also need to insert the recently created property in the actuator.
What we're doing here is adding 1 to the property 60 times per second. So when one second has passed, the property is 60. If your video is let's say 10 seconds exact, we know that we have 10 x 60 = 600 refreshes till the end of the video.
Add a 'property' sensor and set the mode to 'more than' (equal could work too, but anything causing blender to skip a refresh could make it fail). The value should be set to 600 in the case of 10 seconds and of course we are going to use the same property again. Onto this sensor we need to add an 'and' controller and a 'Scene' actuator. Set the mode to 'Set scene' and insert the scene that you use for your game. Now, once the video has ended, your game will start.